Discrete and Digital Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 81

Fig. 2.12 An example of a
recursive digital system
(leaky integrator)

x ( n ) y ( n )

z −1

b

Example (2)) The difference equation that describes the recursive system in Fig.
( 2.12 ) is given by:

y ð n Þ¼ x ð n Þþ by ð n 1 Þ; ð 2 : 14 Þ

where, b is a multiplicative constant (gain). Taking the ZT of both sides of ( 2.14 )
yields:

Y ð z Þ¼ X ð z Þþ bz 1 Y ð z Þ:

Re-arranging terms leads to

½ 1 bz 1 Y ð z Þ¼ X ð z Þ:
)H ð z Þ¼ Y ð z Þ

:

1 bz 1 ¼ z

1

X ð z Þ ¼ z b

From Tables (z-Transform Pairs), the IZT is h(n) = b n u(n).

The system frequency response is:

H ð x Þ¼ H ð z Þj z ¼ exp ð jxT s Þ
¼ e jxT s

e jxT s b ¼ e j2pfT s e j2pfT s b ¼ e j2pf = fs

e j2pf = f s b

¼ e j2pm

e j2pm b ¼ e jX e jX b

where m = f/f s and X ¼ 2pm are the normalized cyclic frequency and normalized
angular frequency, respectively. The magnitude and phase responses are:

H ð x Þj¼j H ð e jxT s Þj¼ j e jxT s j

1
j½ cos ð xT s Þ b j sin ð xT s Þj j e jxT s b j ¼

1
½ cos ð xT s Þ b 2 ½ sin ð xT s Þ 2

¼ q \H ð x Þ

b sin ð x T s Þ
1 b cos ð x T s Þ

¼ \H ð e jxT s Þ¼ tan 1

As a special case, let b = 0.5 and f s = 1 Hz (i.e., T s = 1 s). At f = 0.1 Hz (i.e.,
x = 2p(0.1) = 0.6283rad/s), the magnitude response is j H ð e jxT s Þj¼ 1 : 506 and